Poisson Window

The Poisson window (or more generically exponential window) can be written

$$w_P(n) = w_R(n)e^{- \alpha \frac{\vert n\vert}{ \frac{M-1}{2} }}$$ (4.33)

where $\alpha$ determines the time constant $\tau$ :

$$\frac{\tau}{T} = \frac{M-1}{2\alpha}\quad\hbox{samples}$$ (4.34)

where $T$ denotes the sampling interval in seconds.

Figure 3.19: The Poisson (exponential) window.

The Poisson window is plotted in Fig.3.19. In the $z$ plane, the Poisson window has the effect of radially contracting the unit circle. Consider an infinitely long Poisson window (no truncation by a rectangular window $w_R$ ) applied to a causal signal $h(n)$ having $z$ transform $H(z)$ :

$$\begin{eqnarray*} H_P(z) &=& \sum_{n=0}^\infty [w(n)h(n)] z^{-n} \\ &=& \sum_{n=0}^\infty \left[h(n) e^{- \frac{ \alpha n}{ M/2 }}\right] z^{-n} \qquad\hbox{(let $r\isdef e^{-\frac{\alpha}{ M/2 }}$)}\\ &=& \sum_{n=0}^\infty h(n) z^{-n} r^{n} = \sum_{n=0}^\infty h(n) (z/r)^{-n} \\ &=& H\left(\frac{z}{r}\right) \end{eqnarray*}$$

Thus, the unit-circle response is moved to $\vert z\vert=1/r$ . This means, for example, that marginally stable poles in $H(z)$ now decay as $r^{-n}=e^{-\alpha n/(M/2)}$ in $H(zr)$ .

The effect of this radial $z$ -plane contraction is shown in Fig.3.20.

Figure 3.20: Radial contraction of the unit circle in the $z$ plane by the Poisson window.

The Poisson window can be useful for impulse-response modeling by poles and/or zeros (``system identification''). In such applications, the window length is best chosen to include substantially all of the impulse-response data.